Mathematics is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
The knight's tour is a mathematical chess problem in which the piece called the knight is to visit each square on an otherwise empty chess board exactly once, using only legal moves. It is a special case of the more general Hamiltonian path problem in graph theory. (A closely related non-Hamiltonian problem is that of the longest uncrossed knight's path.) The tour is called closed if the knight ends on a square from which it may legally move to its starting square (thereby forming an endless cycle), and open if not. The tour shown in this animation is open (see also a static image of the completed tour). On a standard 8 × 8 board there are 26,534,728,821,064 possible closed tours and 39,183,656,341,959,808 open tours (counting separately any tours that are equivalent by rotation, reflection, or reversing the direction of travel). Although the earliest known solutions to the knight's tour problem date back to the 9th century CE, the first general procedure for completing the knight's tour was Warnsdorff's rule, first described in 1823. The knight's tour was one of many chess puzzles solved by The Turk, a fake chess-playing machine exhibited as an automaton from 1770 to 1854, and exposed in the early 1820s as an elaborate hoax. True chess-playing automatons (i.e., computer programs) appeared in the 1950s, and by 1988 had become sufficiently advanced to win a match against a grandmaster; in 1997, Deep Blue famously became the first computer system to defeat a reigning world champion (Garry Kasparov) in a match under standard tournament time controls. Despite these advances, there is still debate as to whether chess will ever be "solved" as a computer problem (meaning an algorithm will be developed that can never lose a chess match). According to Zermelo's theorem, such an algorithm does exist.
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